A Direct Approach Based on C 2-IULOWA Operator for Group Decision Making with Uncertain Additive Linguistic Preference Relations

With respect to group decision making (GDM) problem with uncertain additive linguistic preference relations (UALPRs), we investigate the efficient aggregation of the uncertain additive linguistic preference information. First, we introduce two measures to assess the consistency level and the consensus level of uncertain additive linguistic preference information, respectively, and study some of their desirable properties. Then, based on both the two measures, we propose a coinduced uncertain linguistic ordered weighted averaging (IULOWA) operator, called the consistency and consensus coinduced uncertain linguistic ordered weighted averaging (C-IULOWA) operator, to aggregate individual uncertain additive linguistic preference information, in which the consistency level and the consensus level synergistically serve as inducing variables and then guide the determination of the associated weights. We have proved the collective uncertain linguistic preference information aggregated by the C-IULOWA operator that can maintain the fundamental properties of preference relation, such as indifference, reciprocity, and transitivity. By using the C-IULOWA operator, we develop a direct GDM approach with UALPRs. Finally, an illustrative example on the selection of chief quality officer is used to demonstrate the effectiveness and rationalitly of the developed approach.


Introduction
In group decision making (GDM) problems, preference relations are commonly used by decision makers (DMs) to express their preference information based on pairwise comparisons of alternatives.Three of the most common types of preference relations are multiplicative preference relations [1], fuzzy preference relations [2][3][4][5][6][7], and linguistic preference relations [8][9][10].Due to the complexity and uncertainty involved in real-world decision problems and the inherent subjective nature of human judgments, linguistic preference relations can be more appropriate for capturing the lack of precision in human behavior than others.Hence, decision making based on linguistic preference relation has attracted considerable research interests over the past decades [11][12][13][14][15][16][17].
However, sometimes, the DMs are willing or able to provide only uncertain (or interval) linguistic information in some particular situations because of time pressure, lack of knowledge or data, and DMs' limited expertise related to the problem domain [18][19][20][21][22][23][24].Xu [18] proposed the concept of uncertain linguistic variable whose value is interval linguistic information and developed the uncertain linguistic ordered weighted averaging (ULOWA) operator and the uncertain linguistic hybrid aggregation (ULHA) operator to deal with multiple attribute group decision making problems.Xu [19] defined the concept of uncertain additive linguistic preference relation (UALPR) and developed a direct approach to GDM with uncertain linguistic averaging (ULA) and uncertain linguistic weighted averaging (ULWA) operators.Xu [20] introduced the concept of uncertain multiplicative linguistic preference relation and the operational laws of uncertain multiplicative linguistic variables and then proposed the uncertain linguistic weighted geometric mean (ULWGM) operator, uncertain linguistic ordered weighted geometric (ULOWG) operator, and induced uncertain linguistic ordered weighted geometric (IULOWG) operator to deal with GDM problems.Gao and Peng [21] presented a novel quantified SWOT analysis methodology with uncertain linguistic preference relations, interval fuzzy preference relations, and interval multiplicative preference relations.Chen et al. [22] developed a new compatibility for the UALPRs and utilized it to determine the optimal weights of experts in the GDM.Chen and Lee [23] presented an interval linguistic labels ordered weighted average (ILLOWA) operator and a consensus measure for autocratic decision making using group recommendations.Peng et al. [24] presented some multigranular uncertain linguistic prioritized aggregation operators to aggregate directly the uncertain linguistic variables whose values come from the linguistic term sets with different granularities and convey the prioritization phenomenon among the aggregated arguments.
In GDM with all kinds of preference relations, consistency and consensus of preference relations play vital roles.The former refers to the capability of DMs to express their preferences without contradiction; the latter reflects the actual levels of agreement amongst all the individual preferences.Thus, in order to derive a scientific and reasonable decision result, measuring consistency level and measuring consensus level of preference relation are indispensable research topics.To date, although a great deal of research has been conducted on the issues [3, 5-7, 9, 13-17], the contributions of consistency and consensus measures of UALPRs are little.
In the process of practical GDM, on the other hand, it is very difficult for all DMs to construct perfect consistent and consensual preference relations due to the uncertainty and complexity of real world and the subjectivity of DMs' judgments.Also, as pointed out by Saaty, improving consistency does not mean getting an answer closer to reallife situation.It only means that the judgments are closer to being logically related than to being randomly chosen [25].Furthermore, forcing improvements in consistency may distort the individual's true answer to some extent.The similar situations also exist in consensus.In this paper, we consider the problem that if the DMs are no longer available after constructing preference relations and if the preference relations are not perfect consistent and consensual, then how to determine the scientific and reasonable decision results with a higher level of consistency and consensus.To do so, we shall first give the definitions of the measures of consistency and consensus of UALPRs, present a consistency and consensus-based coinduced uncertain linguistic OWA operator to aggregate UALPRs in such a way that more importance is placed on the most consistent and consensual preference information, and then develop a GDM approach with uncertain linguistic preference information.
The rest of this paper is set out as follows.Section 2 briefly introduces relevant concepts of uncertain linguistic preference information.In Section 3 we define the consistency measure and the consensus measure to assess the consistency level and consensus level of uncertain linguistic preference information, respectively, and study some of their properties.On the basis of the two measures, we present a consistency and consensus co-induced uncertain linguistic OWA operator and discuss some of its desirable properties in Section 4. In Section 5, based on the proposed operator, we develop a direct GDM approach.Section 6 demonstrates the effectiveness and rationally of the developed method with an illustrative example on the selection of chief quality officer.Section 7 concludes this paper.

Preliminaries
2.1.Linguistic Variables.Let  = {  |  = 0, 1, 2, . . ., } be a linguistic term set with odd cardinality, which satisfies the following characteristics: (1) the set is ordered: (2) there is the reciprocal operator Neg(  ) =   such as  +  = .For example,  can be defined as To preserve all the given information, Xu [12][13][14] extended the discrete term set  to a continuous term set  = {  |  ∈ [0, ]}, where  ( > ) is a sufficiently large positive integer.If   ∈ , then we call   the original term otherwise, we call   the virtual term.In general, the decision maker uses the original linguistic terms to evaluate alternatives, and the virtual ones can only appear in operation.
is called the deviation between l and its estimated ones  l  = l , 1 ≤  ≤ ,  ̸ = , .
Clearly, when Δ l = (0, 0), then there is no inconsistency at all between l and the rest of preference values in the UALPR, and the higher is Δ l , the more inconsistent is l with respect to the rest of information in the UALPR.
Using the deviation Δ l , we define a consistency measure to assess the consistent level of l .Definition 5. Let L = ( l ) × be an UALPR; then is called the consistency measure of l .
This consistency measure has a definite physical implication and reflects the consistency level between the preference value l and the rest of the preference values in the UALPR.Clearly, when c  = (1, 1), then there is no inconsistency at all between l and the other preference values, and the lower is the value of c  , the more inconsistent is l with respect to the rest of information. Proof.Since then Similarly, we can obtain c   = c   .This completes the above proof.
In particular, if l reduced to   , then the consistency level of   has property   =   .Furthermore, we can obtain the consistency level of the whole UALPR by applying the arithmetic mean.Definition 7. Let L = ( l ) × be an UALPR; then is called the consistency measure of L.
If C = (1, 1), then the UALPR L is fully consistent; otherwise, the lower  t is the more inconsistent L is.
Proof.On has By Theorem 8, we find that the consistency level of the whole UALPR is certain.

Consensus Measure.
Using anonymous ways of the definitions of consistency measure, we define a consensus measure to assess the level of agreement between the individual preference value and the collective ones.
In particular, if UALPRs l reduced to   , then the consensus level of   has property   =   .
Proof.On has By Theorem 13, we find that the consensus level of the whole UALPR is also certain.

Aggregation of Uncertain Linguistic Preference Information Based on Consistency and Consensus
To obtain collective preference information by combining the individual ones, an aggregation operator is needed.The ordered weighted averaging (OWA) and weighted arithmetic averaging (WAA) are the most common aggregation operators.The induced OWA (IOWA) operator provided by Yager and Filev [30] is a more general type of the OWA operator, in which the ordering of the arguments is induced by the order inducing variables, rather than the values of the arguments.
The IOWA operator can be defined as follows.
Definition 14.An IOWA operator of dimension  is a mapping IOWA :   →  can be defined as follows: where  = ( 1 ,  2 , . . .,   )  is an associated weighting vector, such that 0 ≤   ≤ 1, ∑  =1   = 1;  () is the   value of the OWA pair ⟨  ,   ⟩ having the th largest   ;   in ⟨  ,   ⟩ is referred to as the order inducing variable and   as the argument variable.
Note that ( 22) can be equivalently written as where  = where  = ( 1 ,  2 , . . .,   ) is an associated weighting vector, such that 0 ≤   ≤ 1, ∑  =1   = 1; s  is the s value of the OWA pair ⟨  , s ⟩ having theth largest   ⋅   in ⟨  , s ⟩ is referred to as the order inducing variable and s as the argument variable.
The key issues in the IOWA operator theory are to determine the order inducing variables and then derive the associated weights based on them, but quite few studies have been conducted on the issues.Recently, Chiclana et al. [32] introduced a consistency-induced OWA (C-IOWA) operator, in which the ordering of the preference value is based upon its consistency, and it is further studied in [6,7,26,28,29].Wu et al. [17] presented a compatibility index ILOWG (CI-ILOWG) operator, which induces the order of argument values by the compatibility index of experts they also verified that the CI-ILOWG operator guarantees the compatibility degree is at least as good as the arithmetic mean of all the individual compatibility degrees.As aforementioned, both consensus and consistency should be considered simultaneously in any rational GDM.To more effectively relieve the influence some individuals may assign lower consistent and lower centralized preference values to their preferred or repugnant objects, it is very reasonable and meaningful that by assigning low weights to those "inconsistent" or "decentralized" opinions in preference aggregation procedure.Based on the discussions, we here propose a consistency and consensus co-induced uncertain linguistic ordered weighted averaging (C 2 -IULOWA) operator to aggregate individual preference information in such a way that the greater weight is given the most consistent and consensual preference information.
(1) The inducing variables of the C 2 -IULOWA operator are characterized by the consistency measure and the consensus measure, which synergistically reflect the consensus level and the consistency level of UALPRs and participate to guide the determination of associated weights.
(2) The C 2 -IULOWA operator generalizes the consensusinduced uncertain linguistic OWA operators and consistency-induced uncertain linguistic OWA operators, which comprehensively merge the information associated with the properties of preference relation in the preference aggregation.
Moreover, it is easy to verify that the C 2 -IULOWA operator is monotonic, commutative, bounded, and idempotent; the proofs of them are omitted here because they are trivial.
In the following, we study the rationality of the C 2 -IULOWA operator that means the aggregated uncertain linguistic preference relations maintain the essential properties of preference relations such as the indifference, the reciprocity, and the consistency properties.
Since the C 2 -IULOWA operator is idempotent, the indifference property is obvious and its proofs are omitted here.Now we verify the reciprocity property.
Similarly, we can verify l() then we have Similarly, we can verify l() Hence, L() is also consistent, which verifies the consistency property.
An important step in the process of aggregating preference relations is to determine the associated weight of each preference relation or preference value.Usually, the associated weights are determined by a linguistic quantifier  as [32] with () =  1/2 meaning the quantifier guiding this aggregation to be "most" [4,6,32].It can easily be shown that using this the   satisfy the conditions   ∈ [0, 1] and A further important aspect of induced type aggregation operator is the fact that ties in the order inducing variables, that is, when two or more inducing variables have exactly the same value.In order to solve this problem, we recommend adopting Yager's policy [33], is, that using an average of the components of weights over the tied constituents and leaving the arguments alone.

Group Decision Making Based on C 2 -IULOWA Operator
In the procedure of GDM, preference relations are usually composed by two phases [4,[6][7][8][9][10]: the aggregation phase and the exploitation phase, and two distinct choice processes are known: one is the direct way, which draws decision conclusion directly from individual preference relations and aggregates those individual choices, and the other is indirect way, which first aggregates individual linguistic preference relations into a collective preference relation and then draws decision conclusion; they can be described as follows: Herrera et al. [10] and Delgado et al. [11] developed a direct GDM approach with linguistic preference relation, which ranks or selects the alternative(s) according to linguistic nondominance degrees.The approach can be briefly described as follows.
(1) For each linguistic preference relation of each expert,  () , find its respective linguistic strict preference relation, (2) For each linguistic strict preference relation of each expert,   () , determine the individual linguistic non-dominance degree of each alternative   , called IND ()  , according to linguistic OWA operator: (3) For each alternative   , calculate the social linguistic non-dominance degree, called SND  , as follows: (4) Rank or select the alternatives with respect to SND  ( = 1, 2, . . ., ).
Here, we extend those definitions to rank or select the alternative(s) for UALPRs.To do so, we give the following definitions.
Definition 19.Let L = ( l ) × be an UALPR; then the uncertain linguistic dominance relation in terms of L is defined as The linguistic dominance relation is used to measure the "degree of preference" of   over   that exceeds the "degree of preference" of   over   .Definition 20.Let L = ( l ) × be an UALPR; then the uncertain linguistic non-dominance relations are defined as: Since each uncertain linguistic preference value associates with its consistency level and consensus level, at the same time, from Theorem 6,  ; similar ideas can also be found in many existing references [26,27,34,35].Then motivated by [36,37], we give alternative uncertain linguistic nondominance degrees of alternatives by utilizing C 2 -IUOWA operator: Finally, we obtain collective uncertain linguistic nondominance degrees of alternatives by utilizing the C 2 -IULOWA operator again: () , C () , L ()   ⟩ ,  = 1, . . ., ) ,  = 1, 2, . . ., . (42) To rank the uncertain linguistic non-dominance degrees, we adopt to the degree of possibility of uncertain linguistic variables ã ≥ ã [18][19][20].
Now we can rank or select alternatives in descending order in accordance with   ,  ∈ {1, . . ., }.
The main steps of the proposed approach are summarized as follows.

Numerical Examples
In this section, we consider a group decision making problem that concerns the evaluation and selection of chief quality officer to demonstrate the details of the proposed method.
Over the last decades, organizational competition is shifting from price to quality in many industries.The chief quality officer (CQO), similar to the CEO, undertakes overall responsibility of the quality work of enterprise.Hence, the selection of the most appropriate CQO is naturally one of the key factors for an enterprise's survival.A high-tech enterprise wants to introduce a CQO according to a main criterion of the leadership of quality improvements.After preliminary screening, four alternatives   ( = 1, 2, 3, 4) have remained in the candidate list.Three experts   ( = 1, 2, 3) from a committee act as decision makers; the decision makers compare these four alternatives with respect to the criterion of the leadership of quality improvements by using the linguistic term set  = { 0 = extremely low,  1 = very low, To get the most desirable CQO, the following steps are involved.
Step 1.Three decision makers ( 1 ,  2 , and  3 ) provide their preference on alternatives and form UALPRs as follows, respectively, ) . (45) Step 2. Calculate the consistency levels of preference values by using (8); the consistency levels of preference values provided by three decision makers are listed in Tables 1, 2, and 3, respectively.
According to the results from Tables 1, 2, and 3, calculate the consistency level of preference relations by utilizing (12); the results are as follows: Step 3. Calculate the consensus measures of preference values by utilizing (16).The consensus levels of preference values provided by three experts are listed in Tables 4, 5, and 6, respectively.(52) Step 8. Obtain the ranking result of the alternatives according to the magnitude of collective non-dominance degrees by utilizing (43).The ranking result of the four alternatives is  2 ≻  4 ≻  3 ≻  1 , and then the most appropriate CQO is  2 .
In order to validate the proposed method, in what follows, we use the Ct-IULOWA and the Ca-IULOWA operators for the same decision structure.The final results derived by C 2 -IULOWA, Ct-IULOWA, and Ca-IULOWA operators are shown in Table 7, and they are represented graphically in Figure 1.
Obviously, the ranking results derived by C 2 -IULOWA, Ct-IULOWA, and Ca-IULOWA operators all are  2 ≻  4 ≻  3 ≻  1 ; the situations verify each other; the results derived from the C 2 -IULOWA, Ct-IULOWA, and Ca-IULOWA operators are feasible and effective.The reasons of the differences of the final results are intuitive; that is, as discussed above, the Ct-IULOWA operator focuses solely on the consistency and ignores the consensus of the uncertain linguistic preference information, while the Ca-IULOWA operator focuses only on the consensus and ignores the consistency of the uncertain linguistic preference information.The C 2 -IULOWA operator comprehensively considers both the consistency and consensus of the uncertain linguistic preference information.Hence, the results derived by C 2 -IULOWA operator are more feasible and effective.

Conclusion
In group decision making with uncertain linguistic preference relations, the aggregation of preference information plays important roles in reaching a reasonable decision result.
Considering the fact that when carrying out rational decision making, the consensual or consistent information is more appropriate than the inconsistent or decentralized ones, in this paper, we investigated the aggregation of uncertain linguistic preference information by fusing its consensus and consistent information.To do so, we first proposed the consistency and consensus measures, to assess the consistency level and consensus level of uncertain linguistic preference relations and preference values of them, and studied some of their desirable properties.Based on both consistency level and consensus level, we proposed a co-induced uncertain linguistic OWA operator, namely, C 2 -IULOWA operator, to aggregate uncertain linguistic preference information, in which consistency level and consensus level synergistically serve as inducing variables and participate in guiding the determination of associated weights in the preference aggregation process.We have verified that the C 2 -IULOWA operator is able to maintain the indifference, reciprocity, and consistency properties of uncertain linguistic preference relations after aggregation is carried out.For applying the C 2 -IULOWA operator to group decision making with uncertain linguistic preference relations, we developed a direct group decision making approach.The approach is very suitable for the situations with time pressure and decision makers unwilling to revise their initial judgments, in which the inconsistent or decentralized information would be still retained and yet it is considered to be less important for ranking the alternatives.The proposed approach is effective and feasible just as shown in the illustrative example.It is expected that the proposed approach can be applied to the fields of quality management and mobile business.
L(  ,   ) indicates the preference degree of the alternative   over   and is interpreted as   is l as   , l  and l  are the lower and upper limits of l , respectively, l  , l  ∈ [ 0 ,  1 , . . .,   ].In particular, if l  = l  , ∀,  ∈ {1, . . ., }, then l is reduced to a linguistic preference value   .

Table 1 :
Consistency levels of preference values provided by  1 .

Table 2 :
Consistency levels of preference values provided by  2 .

Table 5 :
Consensus levels of preference values provided by  2 .

Table 6 :
Consensus levels of preference values provided by  3 .